Moments of randomly stopped $U$-statistics

Tze Leung Lai; Victor H. de la Peña

doi:10.1214/aop/1023481120

October 1997 Moments of randomly stopped $U$-statistics

Tze Leung Lai, Victor H. de la Peña

Ann. Probab. 25(4): 2055-2081 (October 1997). DOI: 10.1214/aop/1023481120

Abstract

In this paper we provide sharp bounds on the $L_p$-norms of randomly stopped $U$-statistics. These bounds consist mainly of decoupling inequalities designed to reduce the level of dependence between the $U$-statistics and the stopping time involved. We apply our results to obtain Wald’s equation for $U$-statistics, moment convergence theorems and asymptotic expansions for the moments of randomly stopped $U$-statistics. The proofs are based on decoupling inequalities, symmetrization techniques, the use of subsequences and induction arguments.

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NEW YORK, NEW YORK 10027 STANFORD, CALIFORNIA 94305 E-MAIL: karola@stat.stanford.eduNEW YORK, NEW YORK 10027 STANFORD, CALIFORNIA 94305 E-MAIL: karola@stat.stanford.edu

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Tze Leung Lai and Victor H. de la Peña "Moments of randomly stopped $U$-statistics," The Annals of Probability 25(4), 2055-2081, (October 1997). https://doi.org/10.1214/aop/1023481120

Published: October 1997

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